Linear and Nonlinear Boundary Conditions for
Wave Propagation Problems
Jan Nordström
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
Jan Nordström, Linear and Nonlinear Boundary Conditions for Wave Propagation Problems,
in Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws: Notes
on Numerical Fluid Mechanics and Multidisciplinary Design, eds Rainer Ansorge , Hester
Bijl , Andreas Meister and Thomas Sonar, Springer, 2013, 283-299, ISBN 978-3-642-332203.
http://dx.doi.org/10.1007/978-3-642-33221-0_17
Copyright: Springer
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-84545
Linear and Nonlinear Boundary Conditions for
Wave Propagation Problems
Jan Nordström
Abstract We discuss linear and nonlinear boundary conditions for wave propagation problems. The concepts of well-posedness and stability are discussed by considering a specific example of a boundary condition occurring in the modeling of
earthquakes. That boundary condition can be formulated in a linear and nonlinear
way and implemented in a characteristic and non-characteristic way. These differences are discussed and the implications and difficulties are pointed out. Numerical
simulations that illustrate the theoretical discussion are presented together with an
application that show that the methodology can be used for practical problems.
1 Introduction
The principles for construction stable and convergent high order finite difference
schemes for linear and nonlinear boundary conditions are discussed in the context
of wave propagation problems in earthquake simulations.
1.1 Recipe for Constructing a Scheme
The first requirement for obtaining a reliable solution is well-posedness, see [6],[16].
A well posed problem is bounded by the data of the problem and has a unique
solution. Uniqueness for linear problems follows more or less directly from the
energy estimate. This is however not the case for nonlinear problems and we will
investigate that in detail below. Existence is motivated by using a minimal number
of boundary conditions. In the rest of the paper we assume that existence is not a
Jan Nordström
Division of Computational Mathematics, Department of Mathematics, Linköping University, 581
83 Linköping Sweden e-mail: [email protected]
1
problem and will not discuss it further. The crucial point in obtaining well-posedness
is the boundary conditions. These will be chosen such that an energy estimate is
obtained with a minimal number of conditions.
Once we have a well-posed problem, it is meaningful to construct a numerical
approximation. We will use high order finite differences on Summation-By-Parts
(SBP) form and impose the boundary conditions weakly using penalty terms. More
details on this productive and well tested technique is given below. For a read-up, see
[3],[11],[13],[14],[20],[19],[21],[15], [2],[5]. A recipe for constructing a stable and
convergent scheme when using the SBP-SAT technique is to choose the so called
penalty parameters such that an energy-estimate is obtained.
For linear problems, the recipe outlined above guarantees that the scheme converges to a reliable solution as the mesh size goes to zero. However, as we shall see
below, this is not always the case for nonlinear boundary conditions. The difference
in analysis due to the linear and nonlinear version of the boundary condition/friction
law is the main topic of this paper.
1.2 Modeling Related to Earthquake Simulations
The material is modeled as linear elastic with frictional sliding occurring on thin internal interfaces. The internal interfaces, or faults, are governed by highly nonlinear
friction laws. The friction laws relate the slip velocities to the tractions acting on the
fault. They constitute a set of nonlinear boundary/interface conditions. The elastic
wave equations govern the wave propagation between the faults. For a read-up on
these problems see [8],[9].
A simplified but realistic model of problems of this sort is given by
v
01
ut = Auy , y ≥ 0, u =
, A=
, u(y, 0) = f (y).
(1)
σ
10
Note that we have both ingoing and outgoing waves at the boundary y = 0 where
the fault is located. At y = 0 we have either a linear friction law σ = λ v with λ
being a constant or, a highly nonlinear friction law of the general form σ = F(v).
The relation between the velocity and stress is visualized in Figure 1.
We conclude this section by making the assumption that all solutions decay as y
increases i.e limy→∞ u = 0. This assumption simplifies the analysis and enables us to
focus on the interesting boundary with the friction law. In the rest of this paper, all
boundary terms are evaluated at y = 0. The boundary terms for large y are neglected.
y
σ(y,t)
v(y,t)
elastic
incident / reflected
wave
rigid
nonlinear
friction fault
Fig. 1 A schematic of a fault with one-sided friction laws. The lower part of the fault is rigid.
2 Analysis
Below we outline the standard recipe for constructing a stable scheme for a linear
problem. The nonlinear boundary condition will force us to slightly modify that.
2.1 Well-posedness
We will consider two different formulations with slightly different characters.
2.1.1 Non Characteristic Formulation
The energy-method applied to (1 ) yields:
Z ∞
2
0
uT ut dy = kukt2 = −2vσ .
To obtain a bounded solution kuk2 ≤ k f k2 , the linear and nonlinear friction laws
must obey
λ ≥ 0, vF(v) ≥ 0
(2)
respectively.
Next we consider uniqueness and start with the linear case. Consider the difference problem for ∆ u = u1 − u2 ,
∆v
∆ ut = A∆ uy , y ≥ 0, ∆ u =
, ∆ u(y, 0) = 0
(3)
∆σ
and the boundary condition ∆ σ = σ1 − σ2 = λ ∆ v. The energy-method yields:
k∆ ukt2 = −2∆ v∆ σ = −λ ∆ v2
(4)
and clearly the first condition in (2) that guarantees a bounded energy also guarantees uniqueness (since we obtain k∆ uk2 ≤ 0 by integrating (4)). We summarize the
result in the following Theorem.
Theorem 1. The problem (1) with the linear boundary condition σ = λ v is well
posed if
λ ≥ 0.
(5)
In the nonlinear case we have ∆ σ = σ1 − σ2 = F(v1 ) − F(v2 ). The energymethod applied to the difference equation (3) yields:
k∆ ukt2 = −2∆ v∆ σ = −F 0 (v)∆ v2
(6)
where the intermediate value theorem has been used and v ∈ (v1 , v2 ). Note that an
additional condition, namely F 0 (v) ≥ 0 must be added on to the second condition
in (2) which lead to an energy estimate. We summarize the result in the following
Theorem.
Theorem 2. The problem (1) with the nonlinear boundary condition σ = F(v) is
well posed if
vF(v) ≥ 0 and F 0 (v) ≥ 0.
(7)
2.1.2 Characteristic Formulation
The characteristic formulation is obtained by diagonalizing (1). The result if
+
w
−1 0
wt = Λ wy , w =
,
Λ
=
, w(y, 0) = h(y).
0 +1
w−
(8)
The relation between the characteristic variables and the standard variables are
σ = (w− + w+ )/2,
v = (w− − w+ )/2.
(9)
The friction laws can now be formulated as reflection relations. In the linear case
we have
w+ = Rw− , R = (λ − 1)/(λ + 1).
(10)
In the nonlinear case we need to be more careful. The nonlinear friction law
implies σ − v = F(v) − v which by the use of (9) lead to
w+ = F(v) − v = H(v) = H((w− − w+ )/2).
(11)
According to the implicit function theorem, (11) uniquely determines the ingoing
characteristic variable w+ in terms of the outgoing w− if
∂
w+ − H((w− − w+ )/2) 6= 0.
+
∂w
Consequently, the condition for a unique solution becomes H 0 (v) 6= −2 or
F 0 (v) 6= −1.
(12)
The nonlinear friction law on reflection form can be derived as follows. We have
F(v) − v
F(v) − v
w− .
w+ = σ − v = F(v) − v =
(σ + v) =
σ +v
F(v) + v
We summarize the result as
w+ = Rw− ,
R = (F(v) − v)/(F(v) + v)) = R((w− − w+ )/2).
(13)
To prove well-posedness in the nonlinear characteristic case we need the weighted
norm
Z ∞
δ 0
T
2
w Bwdy = kwkB B =
, δ > 0.
(14)
01
0
The energy-method using the new norm and the reflection condition yields
(kwk2B )t = −(w− )2 (1 − δ R2 ).
(15)
In the linear case (10) we always have |R| < 1 since λ > 0 and we immediately have
an estimate even for a standard norm with δ = 1.
In the nonlinear case, it is not that simple. However, If |R| ≤ C < ∞ then 0 < δ <
1/C2 guarantees the estimate. We need to show that |R| is bounded. The extreme
value of R = (F(v) − v)/(F(v) + v)) is given by
R0 (v) = 2(F 0 (v)v − F(v))/(F(v) + v)2 = 0
which is zero for F(v) = F 0 (v)v. The extreme value is R = (F 0 (v) − 1)/(F 0 (v) + 1),
and hence condition (12) also leads to an energy-estimate. Note that also the first
condition in (7) is required to keep |R| bounded.
Next we turn to the question of uniqueness. The energy-method on the difference
problem using (14) yields
(kdwk2B )t = −((dw− )2 − δ (dw+ )2 ).
+/−
+/−
where dw+/− = w1 − w2 and dw = (dw+ , dw− )T . In the linear case (10) leads
directly to dw+ = Rdw− and uniqueness after integration in time. We summarize
the result as
Theorem 3. The problem (8) with the linear characteristic boundary condition (10)
is well posed.
In the nonlinear case we need to derive the relation dw+ = dRdw− . By the intermediate value theorem we get
σ1 − σ2 = F(v1 ) − F(v2 ) = F 0 (v)(v1 − v2 ),
v ∈ (v1 , v2 ).
Reformulating in terms of difference variables leads to
dR = (F 0 (v) − 1)/(F 0 (v) + 1),
and consequently, by the same arguments as in the energy-estimate, F 0 (v) 6= −1
leads to uniqueness. We summarize the result as
Theorem 4. The problem (8) with the nonlinear characteristic boundary condition
(13) is well posed if
vF(v) ≥ 0 and F 0 (v) 6= −1.
(16)
The condition (16) is less restrictive than (7) (allows for more general types of friction laws). We also see that the nonlinear case is more complex than the linear case
where condition (5) suffice for all formulations.
2.2 Stability
We will use high order finite difference techniques on Summation-By-Parts (SBP)
form and impose the boundary conditions (the friction laws) weakly using the Simultaneous Approximation Term (SAT) technique.
2.2.1 SBP Operators and Weak Non Characteristic Boundary Conditions
The semi-discrete formulation of (1) with the weakly enforced boundary condition
is:
Ut = P−1 Q ⊗ A U + P−1 e0 ⊗ Σ B (U0 )
(17)
where e0 = (1, 0, · · · , 0)T , ⊗ is the Kronecker product, Σ = (Σ1 , Σ2 ) is the penalty
matrix, U = (U0 , U1 , · · · , UN )T , Ui = (vi , σi )T and
Bs (U0 ) = (1, 1)T [σ0 − F(v0 )] .
(18)
We augment (17),(18) with the initial condition U(0) = f.
Note that we have expressed both the linear and nonlinear standard boundary
condition in the same functional form (σ0 = F(v0 )). We have used a summationby-parts (SBP) difference operator P−1 Q (see [10],[18]) and imposed the boundary
conditions weakly using the Simultaneous Approximation Term (SAT) technique
[3]. The SBP difference operators satisfy the relations
P = PT > 0,
Q + QT = EN − E0 = diag(−1, 0, ...0, 1),
(19)
and hence they mimic integration by parts perfectly. More details on the weak imposition of boundary and interface conditions using the SAT technique will be given
below. For a read-up on this technique see [3],[11],[13],[14],[20],[19],[21],[15].
By multiplying (17) from the left with UT (P ⊗ I) we obtain
d
kUk2h = −2v0 σ0 + 2UT0 Σ B(U0 ).
dt
The choice Σ1 = 1 and Σ2 = 0 leads to
d
kUk2h = −2v0 F(v0 ),
dt
(20)
which is completely similar to the continuous estimates in both the linear and nonlinear case. We summarize the result below.
Theorem 5. The approximation (17) of the problem (1) is stable for both the linear
(5) and nonlinear (7) boundary condition if the penalty coefficients Σ1 = 1 and
Σ2 = 0 are used.
Note that the conditions (5) and (7) that lead to well-posedness in the continuous
case, are necessary for stability.
2.2.2 SBP Operators and Weak Characteristic Boundary Conditions
The semi-discrete formulation of (8) with the weakly enforced characteristic boundary conditions is:
Wt = P−1 Q ⊗ Λ W + P−1 e0 ⊗ Σ B (W0 )
(21)
−
where Wi = w+
i , wi
T
and
−
Bc (U0 ) = (1, 1)T w+
0 − Rw0 .
(22)
We augment (21),(22) with the initial condition W(0) = h. Note that we have expressed both the linear and nonlinear characteristic boundary condition in the same
−
functional form (w+
0 = Rw0 ).
By multiplying (21) from the left with WT (P ⊗ B) where B = diag(δ , 1) gives us
the weighted norm required above for well-posedness. We obtain
d
2
− 2
T
kW(t)k2h = δ (w+
0 ) − (w0 ) + 2W0 Σ Bc (W0 ).
dt
Th choice Σ1 = −δ and Σ2 = 0 leads to the estimate
d
2
2
+
− 2
kW(t)k2h = −(w−
0 ) (1 − δ R ) − δ (w0 − Rw0 ) ,
dt
(23)
which is similar to the continuous one both in the linear and nonlinear case. Note
that a small damping term proportional to the deviation in the boundary conditions
is added on. We summarize the result in the following theorem.
Theorem 6. The approximation (21) of the problem (8) is stable for both the linear
(10) and nonlinear (13) boundary condition if the penalty coefficients Σ1 = −δ and
Σ2 = 0 are used.
Note that the condition (5) and (16) that lead to well-posedness in the continuous
case, are necessary for stability. Note also that in the linear case we always have
|R| < 1 since λ > 0 and just as in the continuous case, we immediately have an
estimate even for a standard norm with δ = 1.
2.3 Convergence for Finite Time
We will derive the error equation and investigate under which requirement the numerical solution converge to the analytic solution.
2.3.1 Weak Non Characteristic Boundary Conditions
By inserting the analytical solution Ū (projected onto the mesh) in (17) and subtracting (17) we obtain the error equation
Et = P−1 Q ⊗ A E + P−1 e0 ⊗ Σ B (E0 ) + Te
(24)
where E = Ū − U, E = (E0 , E1 , · · · , EN )T , Ei = (∆ vi , ∆ σi )T is the error in the numerical solution, Te = O(∆ x p ) is the truncation error and
Bs (E0 ) = (1, 1)T [∆ σ0 − (F(v̄0 ) − F(v0 ))] .
(25)
The initial data is zero (we initiate the numerical solution with the exact initial data
projected onto the grid), i.e. E(0) = 0.
In this paper we assume that the truncation error Te = O(∆ x p ) is uniform in
accuracy. In reality, the accuracy close to the boundaries are lower, see [18]. This
is especially true for diagonal norm P which one needs in many cases for stability
reasons, see for example [14],[17]
By multiplying (24) from the left with ET (P ⊗ I) we obtain
d
kEk2h = −2∆ v0 (F(v̄0 ) − F(v0 )) + 2ET (P ⊗ I)Te ,
dt
(26)
where Σ1 = 1 and Σ2 = 0 have been used. In the linear case the first term is negative
by the fact that condition (5) holds. In the nonlinear case, the intermediate value
theorem in combination with the second condition in (7) leads to the same result.
The negative contribution of the first term in (26) and the standard inequality
2(u, v) ≤ ηkuk2 + (1/η)kvk2
(27)
leads to
d
kEk2h ≤ ηkEk2h + (1/η)kTe k2h .
dt
Time integration of (28) leads to the final accuracy result
kE(T )k2h ≤
eηT
η
Z T
0
e−ηt kTe k2h dt = O(∆ x2p ),
(28)
(29)
which we summarize below.
Theorem 7. The solution of the approximation (17) converges to the solution of the
problem (1) with linear (5) and nonlinear (7) boundary condition if the penalty
coefficients Σ1 = 1 and Σ2 = 0 are used.
2.3.2 Weak Characteristic Boundary Conditions
We proceed in the same way as in the previous section and insert the analytical
solution W̄ projected onto the mesh in (21) and subtract (21) to obtain the error
equation
Et = P−1 Q ⊗ Λ E + P−1 e0 ⊗ Σ B (E0 ) + Te
(30)
− T
where Ei = W̄i − Wi = ∆ w+
, Te = O(∆ x p ) is the truncation error and
i , ∆ wi
−
Bc (E0 ) = (1, 1)T ∆ w+
0 − ∆ (Rw0 ) .
(31)
Precisely as in the previous section E(0) = 0. Note that we have expressed both the
linear and nonlinear characteristic boundary condition in the same functional form
−
(∆ w+
0 = ∆ (Rw0 )).
By multiplying (30) from the left with ET (P ⊗ B) (using the weighted norm) we
obtain
d
T
T
2
− 2
kEk2h = δ (∆ w+
0 ) − (∆ w0 ) + 2E0 Σ Bc (E0 ) + 2E (P ⊗ B)Te .
dt
(32)
The relations (11) and (9) together with the intermediate value theorem leads to
0
0
F +1
F −1
+
−
+
0
+
∆ w0 − ∆ (Rw0 ) = ∆ w0 − H (v̂)∆ v =
∆ w0 −
∆ w−
0 , (33)
2
2
where v̂ ∈ (v̄, v). By inserting Σ1 = −δ , Σ2 = 0 and (33) in (32) we get
d
δ (F 0 − 1)/2
−δ F 0
2
T
E0 + 2ET (P ⊗ B)Te .
kEkh = E0
δ (F 0 − 1)/2
−1
dt
(34)
By computing the eigenvalues of the boundary matrix and demanding them to be
negative, we find that we must satisfy the second condition in (7).
Consequently, the same conditions an in the non-characteristic case are required
also for the characteristic case although it seemed less restrictive for a while. If the
second condition in (7) holds, we can use (27) and (28) and obtain an estimate like
(29) also in the characteristic case. This proves that the solution converges to the
correct solution using both types of boundary conditions. We summarize that in the
following Theorem.
Theorem 8. The solution of the approximation (21) converges to the solution of the
problem (8) with linear (10) and nonlinear (13) boundary condition if the penalty
coefficients Σ1 = −δ and Σ2 = 0 are used. In the nonlinear case, also the condition
(7) must hold.
Note that condition (16) does not suffice for convergence.
2.4 An Error Bound in Time
We will show that under certain reasonable assumptions, the error growth in time is
bounded even for long times, see [1] and in particular [12]. Both the equations (26)
and (34) can be written
d
kEk2h ≤ −2C0 |E0 |2 + 2kEkkTe k,
dt
(35)
where C0 is an appropriate non-zero constant. By expanding the left-hand side as
d
d
2
dt kEkh = 2kEkh dt kEkh we get
d
kEkh ≤ −η(t)kEkh + kTe k,
dt
(36)
where η(t) = C0 |E0 |2 /kEk2h .
For the sake of argument, we assume that η(t) = η = const.
independent of time.
R
In that case we can integrate (36) and obtain kE(T )kh ≤ e−ηT 0T eηt kTe (t)kh dt. The
estimate kTe (t)kh ≤ max0≤t≤T kTe (t)kh = (kTe kh )max leads to the final error bound
kE(T )kh ≤ (kTe kh )max
(1 − e−ηT )
.
η
(37)
In the case of a time-dependent η(t) not much is changed as long as η(t) is nonnegative and monotonically increasing. The conclusion (37) still holds, see [12].
In conclusion we have that both the characteristic and non-characteristic boundary
conditions imposed weakly lead to an error bound in time.
3 Numerical Results
The results presented here are given in [8],[9],[7],[4]. The figures and tables in Sections 3.1-3.2 and Figure 1 are included with kind permission from Springer Science+Business Media B.V.
3.1 Time-integration and Stiffness
The two different formulations (the standard and the characteristic) of the boundary
conditions were tested. The linear friction law was used in Figure 2 for both cases.
Figure 2 show that the spectral radius of non-characteristic formulation increased
1000
h Im λ ( α) /c s
2
1
0
−2
100
− 282
λ nc (0.1)
λ nc (100)
λ c ( α)
−1
− 0.3
− 0.2
− 0.1
h Re λ ( α) /c s
Non-Characteristic
Characteristic
10
0
10.1
1
10
100
Fig. 2 The spectrum (left figure) using the standard (non-characteristic) boundary conditions and
the spectral radius of the two formulations (to the right).
linearly with α and that the characteristic formulation was independent of α. The
characteristic boundary conditions were clearly less stiff and will be used in the
reminder of the paper.
3.2 Accuracy
The formulation is tested using the friction law F(V ) = 100arcsinh(100V ). The
initial condition is qi (0) = 1000 g(yi )(1, 1)T where
!
1 2
1
g(y) = sin (20 π y) exp −
y−
,
2
2
which corresponds to a wave packet moving to the left (into the fault). We compare
weak and strong (injection type) of boundary conditions. Strong implementation
of boundary conditions mean that the solution values at boundaries are identical
(overwritten) to the boundary data. Note, there is no stability proof for the strong
implementation.
3.2.1 Accuracy for Short Times
Time is truncated at tend = 1 and the domain at y∗ = 1 with a homogeneous nonreflecting boundary condition.
N
40
80
160
320
640
1280
2560
5120
Injection Method SAT Method
Rate Estimate Rate Estimate
nd
2
3rd 4th 2nd 3rd 4th
0.5 1.0 1.5 0.4 1.0 1.8
0.7 2.1 1.7 0.8 2.0 1.3
1.7 1.6 2.8 1.8 1.8 3.3
1.0 2.7 2.5 1.0 2.6 2.1
2.1 2.7 3.3 2.2 2.8 3.5
2.1 3.3 4.1 2.1 3.3 4.4
2.2 4.1 4.8 2.2 4.2 5.1
2.4 4.4 4.7 2.4 4.5 4.9
The strong and weak formulations gave approximately the same convergence rates.
3.2.2 Accuracy for Long Times
Now, time is truncated at tend = 200 and the domain again at y∗ = 1 with a completely reflecting boundary condition. As can be seen in Figure 3, the strong bound-
10
INJ3
INJ4
SAT3
SAT4
exact
9
Error
Energy
101
SAT3
SAT4
INJ3
INJ4
100
0
50
100
Time
150
200
8
7
6
0
50
100
Time
150
200
Fig. 3 The long time error growth (left figure) and energy decay (right figure) using the strong and
weak implementation of the characteristic boundary formulation.
ary condition lead to long time error and energy growth while the weak boundary
condition had an error bound and the energy decay were close to the exact one.
3.2.3 The Necessity of Error Bounds for Long Time Calculations
Here we again truncated at time at tend = 200 and the domain at y∗ = 1 with a
completely reflecting boundary condition. As can be seen in Figure 4, the strong
1.6
t = 60
t=0
0.25
0.5
y
0.75
1
frictional fault
t = 120
0
free surface
0
t = 180
free surface
frictional fault
t = 180
t = 120
1.6
t = 60
t=0
0.25
0.5
y
0.75
1
Fig. 4 The long time error growth (left figure) and energy decay (right figure) using the strong and
weak implementation of the characteristic boundary formulation.
boundary condition lead to a completely contaminated solution for long times. The
weak boundary condition lead to a solution with accurate information even after
long times.
3.3 Application to a Subduction Zone Megathrust Earthquake
We now consider a more complex application problem to demonstrate the full potential of the method. Note that the full methodology is presented in [8],[9],[7],[4]. The
problem is motivated by the recent magnitude 9.0 Tohoku-Oki, Japan, megathrust
earthquake and the resulting tsunami. The specific geometry we consider is shown
in Figure 5, and is loosely based on the subduction zone structure in the vicinity of
the Japan trench.
The Pacific Plate is being sub-ducted to the west beneath the North American /
Okhotsk Plate, with relative motion across the plate interface (the fault) occurring
during megathrust earthquakes. The Japanese island of Honshu lies at the left edge
of the domain, and the upper boundary of the entire computational domain is the
seaoor (with the ocean deepening oshore until it reaches a maximum depth of about
7 km at the trench). Slip along the plate interface causes vertical deformation of
island
arc
lower crust
trench
seafloor
upper crust
mantle wedge
oceanic
crust
oceanic
mantle
fault
20 km
slip
direction
100 km
Material Layer c p (km/s) cs (km/s) ρ (kg/m3 )
Upper Crust
6.0
3.5
3400
Lower Crust
6.9
4.0
3400
Mantle Wedge
7.5
4.3
3400
Oceanic Crust
5.5
3.2
2700
Oceanic Mantle
7.0
4.0
2700
Fig. 5 Subduction zone geometry for megathrust earthquake problem, shown with a factor of 5.25
vertical exaggeration. Different materials are signified with different colors. Solid lines indicate
boundaries between the different materials and dotted lines indicate the computational (multi-block
grid) boundaries. The fault is highlighted with a thicker solid red line and the seafloor (traction-free
surface) is represented with a solid green line. All other boundaries are absorbing boundaries. The
accompanying table lists material properties.
the seaoor, causing uplift or subsidence of the overlying water layer. Gravity waves
(tsunamis) occur as the sea surface returns to an equilibrium level.
The east (Pacific Plate) side is idealized with a two-layer model (oceanic crust
and mantle). As the Pacific Plate dives beneath the North American / Okhotsk Plate,
it crosses several material layers (idealized here as upper and lower crust and the
mantle wedge). We do not include an ocean layer in this model, due to the large
impedance contrast between water and rock, and instead approximate the seaoor as
a traction-free surface. The small angle between the seafloor and the fault (about 5◦ )
creates an extremely challenging geometry.
(a)
horizontal particle velocity (m/s) at t = 15 s
hypocentral
P wave
free surface
reflections
0
-2
2
vertical particle velocity (m/s) at t = 15 s
100 km
hypocentral
S wave
(b)
horizontal particle velocity (m/s) at t = 95 s
25 km
-2
0
2
vertical particle velocity (m/s) at t = 95 s
Rayleigh
wave
Fig. 6 (a) Wave field at t = 15 s for the subduction zone megathrust earthquake. The actively
slipping part of the fault lies slightly behind the hypo-central S-wave front. (b) Wave field at t = 95
s shortly after the rupture has reached the trench. Figures are to scale with no vertical exaggeration.
Note different scale bars in (a) and (b).
For this problem, the method must be extended to handle multi-block interfaces,
far-field boundaries, complex initial conditions, realistic friction laws and a multitude of other complex physical considerations. We do not intend to go through this
in this paper, but refer the reader to [9],[7],[4].
We conduct the simulation at two levels of resolution. The low resolution run
has ∼ 5 × 106 grid points (2123 in the ξ -direction and 2316 in the η-direction)
with a minimum grid spacing along the fault, interfaces, and exterior boundaries
of 100 m, and interior minimum and maximum grid spacings hmin = 0.19 m and
hmax = 200 m, respectively, where hmin is defined and hmax is defined similarly. The
time step is ∆t = 1.5625 × 10−4 s, corresponding to an S-wave CFL of 0.35. The
200 s simulation requires 1.28 × 106 time steps. The high resolution run has twice
the grid resolution (∼ 107 grid points and 2.56 × 106 time steps).
6(a) shows the wave-field at 15 s. The relative motion of the North American /
Okhotsk Plate and the Pacific Plate is consistent with the sense of slip indicated in
Figure 5. For the Japan trench, the island arc would be on the North American /
Okhotsk Plate side, approximately 250 km from the trench axis (the intersection of
the fault with the seafloor). 6(b) shows the wave field at 95 s, shortly after the rupture
has reached the trench. The material on the North American / Okhotsk Plate side is
moving rapidly to the right, due to an extreme reduction in normal stress (and thus
fault strength) from wave reflections off the seafloor. The wave field is quite rich in
structure, and includes dispersed Rayleigh waves propagating along the seafloor in
the oceanic layers. These details are lost in more simplified models that simplify the
physics and geometric complexity.
4 Conclusions
Linear and nonlinear boundary conditions for wave propagation problems have been
considered. The concepts of well-posedness and stability were discussed by focusing on a specific example of boundary treatment occurring in the modeling of earthquakes.
The boundary condition were formulated in a linear and nonlinear way and implemented in a characteristic and non-characteristic way. These differences were
discussed and the implications and difficulties were pointed out.
To discretize in space we used summation-by-parts difference operators and imposed the boundary conditions weakly using the Simultaneous Approximation Term
(SAT) technique.
It was found that the same conditions that lead to well-posedness for the noncharacteristic boundary conditions also lead to stability and convergence of the numerical solution. This goes for both the linear and nonlinear boundary condition,
even though the nonlinear case was more complex.
The conditions that lead to well-posedness for the characteristic boundary conditions were different than the ones required for the non-characteristic conditions,
except in the linear case. The conditions in the nonlinear case were less restrictive.
The conditions that lead to well-posedness for characteristic boundary conditions
also lead to stability for the discrete problem, but not to convergence. To guarantee
convergence we needed to sharpen the conditions to the same level as in the noncharacteristic case for the nonlinear case.
Both types of boundary conditions lead to error bounded schemes if implemented
weakly.
Numerical simulations that illustrate the theoretical discussion are presented. We
show that the correct accuracy as well as stability properties both for short and long
times agree with the theoretical predictions. Finally, an application that show that
the methodology can be used for practical problems is discussed.
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